This looks, sounds, and plays pretty well. A few things of note, though:
1) I like having the double jump for this kind of game cause it's great for helping me correct my jumps and can allow me to try and push a little extra distance to maybe try and skip a part if I'm skilled enough. Only problem is, your standard jump is so shallow that you're using the double jump for almost every jump as it can allow for way more height than the standard. For instance, in the boss race. Getting on to the wall should only require one jump to make as it is relatively low but you have to double jump to make it, and in the bit after where you have to dash through two spike gaps, you either do it while falling or you double jump because your normal jump just can't do it. I'd consider buffing the normal jump or remove double jumps entirely as there just isn't much use for it
2) The wall jumps are a little hard to do. You have to stay clinging to the wall and redirect where you're going after you jump rather than with the jump. Having the player have a little time that they cling to the wall when they push to the other direction can allow for a much easier time of using it. In other words, a little coyote time for the wall jumps would be nice.
3) I noticed 2 bugs. One was with the grapple that it can just start moving down sometimes, and if you let go, you shoot down pretty fast. The other was sometimes your player would be "destroyed" but you don't actually die. The sprite just plays the death effect and disappears
All in all though, this was really fun to play. I liked the little story you had put in, the game looks really cool, and the challenges you provided were pretty good and doesn't require exact precision from the player. Good work